Farouki, Rida T., Knez, Marjeta, Vitrih, Vito, and Žagar, Emil

Subjects

Mathematics - Numerical Analysis

Abstract

By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.

Farouki, Rida T, Pelosi, Francesca, and Sampoli, Maria Lucia

Subjects

Information and Computing Sciences, Engineering, Mathematical Sciences, Pythagorean-hodograph curves, Complex polynomials, Control-polygon constraints, Quadratic and quartic equations, Software Engineering, Information and computing sciences, and Mathematical sciences

Abstract

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves.

Farouki, Rida T, Knez, Marjeta, Vitrih, Vito, and Žagar, Emil

Subjects

Pure Mathematics, Mathematical Sciences, Pythagorean-hodograph curves, Pythagorean-normal surfaces, Minimal surfaces, Enneper-Weierstrass parameterization, Plateau's problem, Quaternions, Applied Mathematics, Numerical and Computational Mathematics, Computation Theory and Mathematics, Numerical & Computational Mathematics, Applied mathematics, and Numerical and computational mathematics

Abstract

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.

Built Environment and Design, Design, Pythagorean-hodograph curves, Complex polynomials, Curve approximation, Constrained optimization, Lagrange multipliers, Mechanical Engineering, Design Practice and Management, Design Practice & Management, and Mechanical engineering

Abstract

The problem of identifying the planar Pythagorean-hodograph curve that is “closest” to a given Bézier curve, and has the same end points (or end points and tangents), is considered. The “closeness” measure employed in this context is the root-mean-square magnitude of the differences between pairs of corresponding control points for the two curves. The methodology is developed in the context of quintic PH curves, although it readily generalizes to PH curves of higher degree. Using the complex representation for planar curves, it is shown that this problem can be reduced to the minimization of a quartic penalty function in certain real variables, subject to two quadratic constraints, which can be efficiently solved by the Lagrange multiplier method. By expressing the penalty function and constraints in terms of variables that identify a complex pre-image polynomial, the closest solution is guaranteed to be a PH curve. Several computed examples are used to illustrate implementation of the optimization methodology and typical approximation results that can be obtained.

The International Journal of Advanced Manufacturing Technology. 119(9-10)

Subjects

CNC machine, Backlash, Machine dynamics, Osculating circle, Feedrate modulation, Position error, Mathematical Sciences, Information and Computing Sciences, Engineering, and Industrial Engineering & Automation

Abstract

A methodology for analyzing the influence of gear backlash in the axis drive systems of a Cartesian CNC machine on positional accuracy is developed. The approach is based on solving the machine dynamical equations in the context of an angular dead-zone backlash model and an osculating circle approximation of smooth paths in a neighborhood each path turning point, which admit an essentially exact solution for a P controller. This methodology is the basis for schemes to minimize backlash degradation of positional accuracy through feedrate or path geometry modifications, rather than hardware or controller upgrades (that may be expensive or disruptive for CNC machines in continuous production use). As a preliminary demonstration of the methodology, results are presented from the use of smooth feedrate reductions about each path turning point as a means to suppress positional inaccuracies incurred by gear backlash in CNC machine axis drive systems.

An algorithm for real–time venipuncture needle guidance is described, using an optical coherence tomography (OCT) probe that emits light pulses at fixed angular intervals along a cone, giving accurate distance measurements to points on the blood vessel. Using this data, a method is developed to visually display the blood vessel for needle guidance. A least–squares fit to a general quadric surface, specified by a symmetric matrix, is performed. For a cylindrical blood vessel, this provides an estimate for its orientation, from which its location and radius can be determined. The algorithm is compatible, in efficiency and robustness, with real–time implementation.

Farouki, Rida T, Knez, Marjeta, Vitrih, Vito, and Žagar, Emil

Subjects

Information and Computing Sciences, Engineering, Mathematical Sciences, Pythagorean-hodograph curves, Helical curves, Planar orthogonal projections, Polynomial factorizations, Quaternions, Spatial rotations, Software Engineering, Information and computing sciences, and Mathematical sciences

Abstract

Although the orthogonal projection of a spatial Pythagorean–hodograph (PH) curve on to a plane is not (in general) a planar PH curve, it is possible to construct spatial PH curves so as to ensure that their orthogonal projections on to planes of a prescribed orientation are planar PH curves. The construction employs an analysis of the root structure of the components of the quaternion polynomials that generate spatial PH curves, and it encompasses both helical and non–helical spatial PH curves. An initial characterization for orthogonal projections of spatial PH curves on to the coordinate planes provides the basis for a generalization to projections of arbitrary direction, based on unit quaternion rotation transformations of R3.

Real-time CNC interpolator, Parametric curves, Variable feedrate, Taylor series coefficients, Richardson extrapolation, Feedrate accuracy, Mechanical Engineering, Design Practice and Management, and Design Practice & Management

Abstract

Real-time CNC interpolators achieving a constant or variable feedrate V along a parametric curve r(ξ) are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter ξ. Since the feedrate should be specified as a function of a physically meaningful variable (such as time t, path arc length s, or curvature κ) rather than ξ, successive applications of the differentiation chain rule are necessary to determine Taylor series coefficients beyond the linear term. The closed-form expressions for the higher-order coefficients are increasingly cumbersome to derive and implement, and consequently error-prone. To address this issue, the use of Richardson extrapolation as a simple means to compute rapidly convergent approximations to the higher-order coefficients is investigated herein. The methodology is demonstrated in the context of (1) an arc-length-dependent feedrate for cornering motions; (2) direct real-time offset curve interpolation; and (3) a curvature-dependent feedrate. All of these examples admit simple implementations that circumvent the need for tedious symbolic calculations of higher-order coefficients, and are compatible with real-time controllers with millisecond sampling intervals.

Farouki, Rida T, Knez, Marjeta, Vitrih, Vito, and Žagar, Emil

Subjects

Applied Mathematics, Numerical and Computational Mathematics, Mathematical Sciences, Spatial closed-loop curves, Continuity conditions, Arc length, Pythagorean-hodograph curves, Euler-Rodrigues frame, Tubular surfaces, Computation Theory and Mathematics, Numerical & Computational Mathematics, Applied mathematics, and Numerical and computational mathematics

Abstract

We investigate the problem of constructing spatial C2 closed loops from a single polynomial curve segment r(t), t∈[0,1] with a prescribed arc length S and continuity of the Frenet frame and curvature at the juncture point r(1)=r(0). Adopting canonical coordinates to fix the initial/final point and tangent, a closed-form solution for a two-parameter family of interpolants to the given data can be constructed in terms of degree 7 Pythagorean-hodograph (PH) space curves, and continuity of the torsion is also obtained when one of the parameters is set to zero. The geometrical properties of these closed-loop PH curves are elucidated, and certain symmetry properties and degenerate cases are identified. The two-parameter family of closed-loop C2 PH curves is also used to construct certain swept surfaces and tubular surfaces, and a selection of computed examples is included to illustrate the methodology.

Farouki, Rida T, Pelosi, Francesca, and Sampoli, Maria Lucia

Subjects

Theory Of Computation, Applied Mathematics, Information and Computing Sciences, Numerical and Computational Mathematics, Mathematical Sciences, Clothoid, Cornu spiral, Fresnel integrals, Arc length, Pythagorean-hodograph curves, Geometric Hermite interpolation, Electrical and Electronic Engineering, Numerical & Computational Mathematics, Theory of computation, Applied mathematics, and Numerical and computational mathematics

Abstract

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution.

Information and Computing Sciences, Engineering, Mathematical Sciences, Pythagorean-hodograph curves, Algebraic equations, Hermite interpolation, Complex numbers, Quaternions, Arc length, Software Engineering, Information and computing sciences, and Mathematical sciences

Abstract

A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a “double” solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics.

The International Journal of Advanced Manufacturing Technology. 109(7-8)

Subjects

Model predictive control, Inverse dynamics, CNC machine, System identification, Feedrate, Contour error, Feed error, Mathematical Sciences, Information and Computing Sciences, Engineering, and Industrial Engineering & Automation

Abstract

The use of model predictive control (MPC) as a form of inverse dynamics compensation for multi–axis CNC machines, to subdue the inaccuracies incurred by axis inertia and damping, is investigated by both simulation studies and experimental performance analysis using a 3–axis milling machine governed by an open–architecture software controller. The results indicate that MPC is a viable tool for inverse dynamics compensation with a controller sampling frequency f = 1024 Hz running on a 500-MHz processor, with only modest prediction horizons offering excellent performance in terms of feedrate accuracy and contour error suppression. Unlike inverse dynamics schemes based upon linear time–invariant dynamic models, the MPC scheme provides the flexibility to compensate for nonlinear physical effects such as backlash in the machine axes and hard constraints on axis accelerations imposed by motor torque constraint.